Proceedings of the American Mathematical Society

Author(s): James K. Deveney; David R. Finston; Peter van Rossum
Journal: Proc. Amer. Math. Soc. 132 (2004), 2841-2848.
MSC (2000): Primary 14L30; Secondary 20G20
Posted: June 2, 2004
Retrieve article in: PDF DVI PostScript

Abstract: Every locally trivial action of the additive group of complex numbers on four-dimensional complex affine space that is given by a triangular derivation is conjugate to a translation. A criterion for a proper action on complex affine

-space to be locally trivial is given, along with an example showing that the hypotheses of the criterion are sharp.

1.: A. van den Essen: An algorithm to compute the invariant ring of a $G_{a}$ action on an affine variety, J. Symbolic Computation 16 (1993) 531-555. MR 95c:14064
2.: D. Daigle and G. Freudenberg: Triangular derivations of $\emph{k[X}_{1}\emph{,X}_{2}\emph{,X}_{3}\emph{,X}_{4}\emph{]}$ , J. Algebra 241 (2001) 328-339. MR 2002g:13058
3.: V. I. Danilov: On rings with discrete divisor class group, Math. USSR Sbornik 17 (1972) 228-236. MR 46:5311
4.: J. K. Deveney, D. R. Finston, and M. Gehrke: $G_{a}$ actions on $\mathbf{C}^{n}$ , Comm. Alg. 22 (1994) 4977-4988. MR 95e:14038
5.: J. K. Deveney and D. R. Finston: A proper $G_{a}$ action on $\mathbf{C}^{5}$ which is not locally trivial, Proc. Amer. Math. Soc. 123 (1995) 651-655. MR 95j:14065
6.: J. K. Deveney and D. R. Finston: $G_{a}$ invariants and slices, Comm. Alg. 30 (2002) 1437-1447. MR 2003c:14073
7.: J. K. Deveney and D. R. Finston: Regular $G_{a}$ invariants, Osaka J. Math. 39 (2002) 275-282. MR 2003e:14037
8.: G.-M. Greuel and G. Pfister: Geometric quotients of unipotent group actions II. Singularities (Oberwolfach, 1996), 27-36, Progr. Math. 162, Birkhäuser, Basel, 1998. MR 99k:14078
9.: J. K. Deveney and D. R. Finston: Local triviality of proper $G_{a}$ actions, J. Algebra 221 (1999) 692-704. MR 2001b:14095
10.: D. Wright: On the Jacobian conjecture. Illinois J. Math. 25 (1981) 423-440. MR 83a:12032
11.: H. Popp: Moduli Theory and Classification Theory of Algebraic Varieties, Lecture Notes in Mathematics, no. 620, Springer-Verlag, Berlin, Heidelberg, New York, 1977. MR 57:6024
12.: H. Holmann: Komplexe Räume mit komplexen Transformationsgruppen, Math. Ann. 150 (1963) 327-360. MR 27:776
13.: J. Lipman: Unique factorization in complete local rings, Proc Sympos. Pure Math. 29 (1974) 531-546. MR 51:10325
14.: J. Winkelmann: On free holomorphic C actions on C $^{ \text{n}}$ and homogeneous Stein manifolds, Math. Ann. 286 (1990) 593-612. MR 90k:32094
15.: E. Halanay: Un exemple de inel factorial al carui completat nu este factorial, St. Cerc. Mat. 30 (1978) 495-497. MR 80a:13017
16.: P. van Rossum: Tackling Problems on Affine Space with Locally Nilpotent Derivations on Polynomial Rings, Thesis, Catholic University Nijmegen, 2001.
17.: R. Hartshorne and A. Ogus: On the factoriality of local rings of small embedding codimension, Comm Alg. 1 (1974) 415-437. MR 50:322
18.: D. Mumford: The Red Book of Varieties and Schemes, Lecture Notes in Mathematics, no. 1358, Springer-Verlag, Berlin, Heidelberg, New York, 1999. MR 2001b:14001
19.: S. Milne: Étale Cohomology, Princeton University Press, Princeton, NJ, 1980. MR 81j:14002

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14L30, 20G20

James K. Deveney
Affiliation: Department of Mathematical Sciences, Virginia Commonwealth University, 1015 W. Main St., Richmond, Virginia 23284
Email: jdeveney@atlas.vcu.edu

David R. Finston
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Email: dfinston@nmsu.edu

Peter van Rossum
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Email: petervr@nmsu.edu

DOI: 10.1090/S0002-9939-04-07500-8
PII: S 0002-9939(04)07500-8
Keywords: Additive group, slice, geometric quotient, locally trivial
Received by editor(s): July 25, 2002
Posted: June 2, 2004
Communicated by: Bernd Ulrich
Copyright of article: Copyright 2004, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS